v2007.09.13 - Convex Optimization

E.6. VECTORIZATION INTERPRETATION, 607(A.2) It is a fact that y T Xy is always proportional to a coefficient oforthogonal projection; letting z in formula (1738) become y ∈ R m , thenP 2 =P 1 =yy T /y T y=yy T /‖yy T ‖ 2 (confer (1390)) and formula (1739) becomes〈yy T , X〉〈yy T , yy T 〉 yyT = yT Xyy T yyy Ty T y = yyTy T y X yyTy T y∆= P 1 XP 1 (1750)By (1737), product P 1 XP 1 is the one-dimensional orthogonal projection ofX in isomorphic R m2 on the range of vectorized P 1 because, for rankP 1 =1and P 21 =P 1 ∈ S m (confer (1729))P 1 XP 1 = yT Xyy T y〈 〉yy T yyT yyTy T y = y T y , X y T y = 〈P 1 , X〉 P 1 = 〈P 1 , X〉〈P 1 , P 1 〉 P 1(1751)The coefficient of orthogonal projection 〈P 1 , X〉= y T Xy/(y T y) is also knownas Rayleigh’s quotient. E.11 When P 1 is rank-one symmetric as in (1750),R(vecP 1 XP 1 ) = R(vec P 1 ) in R m2 (1752)andP 1 XP 1 − X ⊥ P 1 in R m2 (1753)E.11 When y becomes the j th eigenvector s j of diagonalizable X , for example, 〈P 1 , X 〉becomes the j th eigenvalue: [144,III]〈P 1 , X 〉| y=sj=s T j( m∑λ i s i wiTi=1s T j s j)s j= λ jSimilarly for y = w j , the j th left-eigenvector,〈P 1 , X 〉| y=wj=w T j( m∑λ i s i wiTi=1w T j w j)w j= λ jA quandary may arise regarding the potential annihilation of the antisymmetric part ofX when s T j Xs j is formed. Were annihilation to occur, it would imply the eigenvalue thusfound came instead from the symmetric part of X . The quandary is resolved recognizingthat diagonalization of real X admits complex eigenvectors; hence, annihilation could onlycome about by forming Re(s H j Xs j) = s H j (X +XT )s j /2 [149,7.1] where (X +X T )/2 isthe symmetric part of X, and s H j denotes the conjugate transpose.